ABJM on ellipsoid and topological strings

It is known that the large N expansion of the partition function in ABJM theory on a three-sphere is completely determined by the topological string on local Hirzebruch surface F 0 $$ {\mathbb{F}}_0 $$ . In this note, we investigate the ABJM partition function on an ellipsoid, which has a conventional deformation parameter b . Using 3d mirror symmetry, we find a remarkable relation between the ellipsoid partition function for b 2 = 3 (or b 2 = 1 / 3) in ABJM theory at k = 1 and a matrix model for the topological string on another CalabiYau threefold, known as local ℙ 2 $$ {\mathrm{\mathbb{P}}}^2 $$ . As in the case of b = 1, we can compute the full large N expansion of the partition function in this case. This is the first example of the complete large N solution in ABJM theory on the squashed sphere. Using the obtained results, we also analyze the supersymmetric Rényi entropy.